Causation and counterfactuals

Let us now consider the way Mackie characterizes sufficient and necessary conditions. Standard definitions of these notions are as follows:
A is a sufficient condition of B iff, if A occurs, B occurs
A is a necessary condition of B iff, if B occurs, A occurs (or, equivalently, if A doesn’t occur, B doesn’t occur).

But these definitions are correct only when A and B are general types of events, and not names of individual objects (in that case the right-hand sides of the equivalences have to be interpreted as general statements: “For all x, if x is A and x occurs, then there is a y such that y is B and y occurs”). If A and B are singular names, the aforementioned definitions wrongly imply that all actual events are sufficient and necessary conditions of each other. Mackie attempts to give a better analysis applicable to singular claims, not general ones. His analysis is presented with the help of following equivalences:

x is a sufficient condition of y iff since x occurred, y occurred
x is a necessary condition of y iff if x had not occurred, y would not have occurred.

Mackie stresses that the conditionals used in each explanans can’t be interpreted as material conditionals. But this creates an immediate problem for the regularity approach, as one of its main assumptions is that causation should be explicated without resorting to modal notions, such as necessity or possibility. Mackie suggests the following interpretations of the non-material conditionals used above. He treats them as “telescoped arguments” in which some premises are omitted. For instance, the statement “If the short circuit had not occurred, there would have been no fire” can be expanded into the statement that there are some true universal propositions which together with true statements about the conditions of the house and together with the supposition that the short circuit did not occur logically imply that there was no fire. A similar analysis can be given for the statement “Since x occurred, y occurred”.


One of the most serious challenges for any account of causation is presented by the so-called redundant causation. There are two main types of redundant causation: overdetermination and pre-emption. Overdetermination occurs when there is more than one acting cause, each of which is sufficient for the effect to occur. An example can be an execution by a firing squad, in which each bullet causes a lethal injury. Is each individual shot a cause of the death of the condemned man? In Mackie’s approach the answer is negative, because one of his characteristics of causation is that no alternative sufficient conditions are present (a cause is necessary post factum for the effect). Only the disjunction of all shots is necessary in this sense.

The case of pre-emption can be described using the following example. Two children, Billy and Suzie, are throwing stones at a bottle. Seeing that Suzie has thrown her stone and shattered the bottle, Billy does not hurl his stone, but if Suzie had not thrown, Billy would have thrown his stone. We call this pre-emption, because Suzie’s throw pre-empts Billy’s action which would otherwise have taken place. Any reasonable theory of causation should imply that Suzie’s throw was the actual cause of the shattering. But wasn’t her throw unnecessary, given that Billy was present as a backup? Are the conditions imposed by Mackie satisfied? It turns out that they are, in spite of our initial worries. According to Mackie’s definition, there has to be a set of conditions X such that together with Suzie’s throw (let’s call it A) they would constitute a sufficient condition for the shattering, and moreover no alternative sets of sufficient conditions can be present. We can reasonably believe that without A Bill’s throw together with its conditions would create a different sufficient condition for the shattering of the bottle, but this set is not present at the time of Suzie’s throw. Suzie’s throw is still a necessary part of its own set of conditions – without it this set would not be sufficient, although a different one would be. So Mackie’s definition gives the right answer in the case of pre-emption.

David Lewis has noted that all regularity theories of causation, including Mackie’s, have problems with distinguishing causes from effects, and direct causal links from correlations arising due to a common cause. If only events of type A can cause B, we can say that a given event B is a sufficient condition (or part of a sufficient condition) of A, and hence B is wrongly classified as a cause of A. Even if we artificially exclude this possibility by stipulating that a cause has to be earlier than its effect, still the problem remains. Suppose that an event A causes B and C in succession, and that B can be created only by events of type A. In such a case B is a sufficient condition (or, as in Mackie’s conception, a necessary part of a sufficient condition) of A, and A is in turn a sufficient condition of C, hence B comes out to be a cause of C.

Lewis suggests an alternative account of causation: a counterfactual analysis. The simplest version of such an analysis (the so-called naive counterfactual analysis) is as follows: x is a cause of y iff if x had not occurred, y would not have occurred. But this definition is in need of serious corrections. For instance, suppose that I shut the door by slamming it. If I hadn’t shut the door, I wouldn’t have slammed it (we assume that each slamming shuts the door), but the shutting of the door is not a cause of its slamming. Similarly, if I hadn’t written the letter “L”, I would not have written the word “Lewis”, but the first did not cause the second. To eliminate these counterexamples, we have to assume that x and y are distinct events (i.e. they are not identical, nor one is part of the other). But in order to further advance the counterfactual analysis, we have to understand better the meaning of counterfactual conditionals.

Counterfactual conditionals can be generally presented as statements of the form “If it were (had been) the case that p, then it would be (have been) the case that q”. It is well known that sentences of that form are not truth-functional, i.e. the truth value of the entire complex statement is not determined by the truth values of its components. To see this, it suffices to compare the following two statements: “If Rodin’s sculpture ‘The Thinker’ were made out of wood, it would float” and “If Rodin’s sculpture ‘The Thinker’ were made out of wood, it would fly”. In both cases the antecedent and the consequent are false, and yet the first conditional is true, while the second false. The counterfactual connective is considered to be a modal one, and it receives an interpretation in terms of possible worlds, similar to that of necessity and possibility. The most commonly accepted analysis stipulates that the conditional “If it were p, then it would be q” is true if and only if q is true in all possible worlds in which p is true and which are closest to the actual world of all p-worlds. Thus the statement “If I threw a stone at a window, it would shatter” is true if in the possible worlds in which I throw the stone and which otherwise are as similar to the actual world as the truth of the antecedent allows, the window shatters. And this is what we expect to get, because in such worlds the laws of nature and the properties of materials such as glass and rock will be the same as in our world. On the other hand, if we considered a far away world in which glass is tougher than rock, the consequent would not be true. This shows that for a counterfactual to be true, the consequent does not have to be true in all worlds in which the antecedent is true (this truth condition defines the so-called strict conditional).

Counterfactual conditionals follow a slightly different logic than material conditionals or strict conditionals. Let us consider the three following logical laws: the law of the strengthening of the antecedent, the law of transposition and the law of transitivity. The first one states that if p implies q, then the conjunction p and r also implies q. But consider the following example: If someone shot a gun pointed at my chest, I would be dead, but if someone shot at me and I was wearing a bulletproof vest, I would survive. Given that in the actual world I am not wearing a bulletproof vest now, both statements seem to be true, which shows that the law of the strengthening of the antecedent is violated for counterfactual conditionals. The reason behind this is that we choose different possible worlds to evaluate both statements: the first one is the world where I am being fired at, but I am not wearing any protection, and the second one is the one in which I am protected by a bulletproof vest.

The law of transposition states that if p implies q, not-q implies not-p. A counterexample to this law is as follows: it is true that if I didn’t come to my lecture, the building in which I am lecturing would still stand. But from this it does not follow that if the building collapsed (for instance because of an earthquake), I would still come to the lecture. Finally, the law of transitivity prescribes that if p implies q, and q implies r, then p implies r. We already presented a case which violates this law when we discussed the problem of the transitivity of the causal relation. In this case p = the bomb was not planted, q = the bomb was not defused, r = the politician was assassinated. We can also note that the violation of transitivity follows directly from the violation of the strengthening of the antecedent, given that it is a logical truth that if it were p and r, then it would be p. Now we can choose p, q and r such that it is true that if it were p, it would be q, but it’s false that if it were p and r, then it would be q, and the transitivity is violated.

Reading:

E.J. Lowe, Chapter 8 "Counterfactual conditionals", pp. 137-154, A Survey of Metaphysics.

Regularity theories of causation

Having rejected the postulate of the necessity of causal links, Hume replaces it with the condition of regularity. Paraphrasing his words, if x is a cause of y, each event that is similar to x is followed by an event similar to y. Thus, the complete definition of causation can look like this: x is a cause of y iff x is spatiotemporally contiguous with y, x temporally precedes y, and for all x’, if x’ is similar to x, then there is a y’ such that y’ is similar to y and y’ follows x’. One of the main difficulties with this definition is the notorious vagueness of the notion of similarity between events. If we interpret similarity as being identical in some respect, then it is plausible that every object is similar to every other object, and as a consequence the condition of regularity can never be satisfied. On the other hand, similarity conceived as identity in all respects (qualitative identity) collapses into numerical identity (given the PII), and therefore the condition of regularity reduces to the previous two conditions (contiguity and temporal precedence). The main challenge to Hume’s analysis is to characterize the notion of a relevant aspect with respect to which the similarity between events is interpreted. But even if this can be done, Hume’s analysis is open to serious criticism. Critics point out that there are cases of regular successions of events without a causal link. Days regularly follow nights, and yet there is no causal relation between the two. Similarly, if according to the timetable after the departure of train A train B regularly arrives at the station, this does not indicate that one causes the other. Note that typically the regular but not causal correlations can be explained with the help of a common cause (the succession of day and night is explained by the rotation of the earth, and the timetable acts as the common cause of both the departure of train A and the arrival of train B). So it can be concluded that Hume’s regularity approach has difficulties with distinguishing between direct causal regularities and regularities arising from a common cause.

Another group of counterexamples to the Humean analysis contains cases of causal links where no regularity is present. It is sometimes argued that causal links between unique events (for instance the Big Bang) don’t satisfy the requirement of regularity. Actually, they satisfy it but trivially, making all spatiotemporally conjoined unique events causally connected with each other. But it is unquestionable that in real life we make numerous causal claims where there is no underlying regularity. We say that the failure of brakes was a cause of the car crash, although not every failure of that sort leads to a crash. This clearly shows that Hume’s analysis is in need of serious corrections.

Neo-Humean approaches to causation try to eliminate some of its weak points. One of them is the nomological conception. According to it, an event x of type A is a cause of an event y of type B iff x and y occur in conditions C and there is a law of nature according to which if an event of type A happens in conditions C, an event of type B occurs. There are two main differences between this approach and Hume’s regularity account: the presence of background conditions C and the reference to laws of nature. The role of laws of nature is to eliminate accidental regularities, while the presence of conditions C should take care of the problem of the apparent lack of regularity of some causal links (the failure of brakes leads to an accident only in certain specific circumstances). But these additions to the regularity approach bring new problems. What are laws? Hume himself insisted that laws are nothing over and above mere regularities, but if that’s the case the addition of laws to the definition of causation does not constitute an improvement with respect to the old, Humean version. The introduction of conditions C creates a different problem. By a crafty selection of appropriate “conditions” we can make virtually any succession of events a causal one. It may be claimed, for instance, that the snapping of my fingers causes the lights to go out if we include in the appropriate conditions that someone turns off the switch at the same moment. Yet another objection is that there are laws which are not causal. Pascal’s law states that a gas exerts equal pressure in all directions, but it is not correct to say that the fact that the pressure in one direction equals p is caused by the fact that the pressure in the opposite direction is also p. Finally, most laws of physics are symmetric in time, but from this it does not follow that backward causation is a common fact.

One of the most sophisticated versions of the regularity approach is the conception of causation proposed by J.L. Mackie. Mackie notes that the notion of a cause is closely related to sufficient and necessary conditions, but a cause of x cannot be simply defined as a necessary or a sufficient condition of x. Let us consider Mackie’s example with a fire of a house being caused by a short circuit. The short circuit by itself is not sufficient for the house to burn down; other conditions have to be present, such as the presence of inflammable materials, of oxygen, the absence of automatic sprinklers and smoke detectors, etc. Generally speaking the short circuit is not necessary for the fire either, for fires can start in many different ways, for instance after a strike of a lightning bolt. But in the actual conditions the short circuit was necessary, because without it the conditions themselves would not have created the fire. (Mackie speaks in this case about a necessary condition post factum.) The short circuit is an INUS condition for the occurrence of the fire, where the acronym INUS stands for an Insufficient but Necessary part of an Unnecessary but Sufficient condition. A more precise definition of an INUS condition is as follows: A is an INUS condition for B iff there are conditions X and Y such that (AX or Y) is a necessary and sufficient condition of B, but neither A nor X is a sufficient condition of B. In our example A is the short circuit, B is the fire, X refers to all the conditions which together with A were sufficient for the fire to occur, and Y stands for a disjunction of all alternatives ways of starting a fire. In most cases a cause of an occurrence B is an INUS condition of B such that it occurred, and no alternative conditions Y were present (however, Mackie admits the possibility that a cause itself may be sufficient for its effect, or even sufficient and necessary – he refers to all such cases jointly as “at least” INUS conditions of B).

It has to be noted that some further restrictions on the causal condition A have to be introduced, otherwise Mackie’s analysis will lead to obviously incorrect conclusions. To see this, let us use the letter C to abbreviate the complete sufficient condition of B that was actually present, and let us select any fact S irrelevant to the occurrence of B that happened simultaneously with C (for instance the fact that when the fire started somebody walked past the house whistling “Ode to joy”). The formula [(S or C) and (not-S or C)] is logically equivalent to C, and hence if there is a condition Y such that (C or Y) is a necessary and sufficient condition of B, then {[(S or C) and (not-S or C)] or Y} is a necessary and sufficient condition of B too. But now observe that the condition (S or C) satisfies the requirement for an INUS condition of B. (S or C) is not sufficient for B, nor is (not-S or C), and yet their conjunction is sufficient (as it is equivalent to C). But it is highly unintuitive to pick the disjunction S or C as a cause of B. To eliminate cases like this it may be suggested that causes should not have the form of disjunctions of simple events.

An interesting element of Mackie’s conception is that he admits that causal claims are always made in a context. In order to account for this fact, he introduces the notion of a causal field. Let us consider as an example the case of a person going down with flu. The answer to the question “What caused this man to contract the flu?” depends on the context. If we consider as the causal field the set of all moments in his life, and ask why he contracted the disease at this moment rather than any other, then the correct answer may be that he was infected by influenza viruses. But we can also select as the causal field the set of all people who came into contact with influenza viruses, and we may be interested in selecting the factor which is responsible for the fact that some of them contracted the disease, while the others did not. Mackie introduces the causal field to his definition of a cause, assuming that the conditions characterizing this field are present when the cause is present.

Reading:

M.J. Loux, Chapter 6 "Causation", pp. 187-203, Metaphysics: A Contemporary Introduction.